5 research outputs found
Absolute Continuity of Semimartingales
We derive equivalent conditions for the (local) absolute continuity of two
laws of semimartingales on random sets. Our result generalizes previous results
for classical semimartingales by replacing a strong uniqueness assumption by a
weaker uniqueness assumption. The main tool is a generalized Girsanov's
theorem, which relates laws of two possibly explosive semimartingales to a
candidate density process. Its proof is based on an extension theorem for
consistent families of probability measures. Moreover, we show that in a
one-dimensional It\^o-diffusion setting our result reproduces the known
deterministic characterizations for (local) absolute continuity. Finally, we
give a Khasminskii-type test for the absolute continuity of multi-dimensional
It\^o-diffusions and derive linear growth conditions for the martingale
property of stochastic exponentials